Actually, it wouldn't be the president, it would be a young missilier who works twelve hour shifts in an underground bunker where he just sits staring at the red button until the call comes in. The message is transmitted in code that the missiliers (two in case one chickens out) have to decode to ensure it's authenticity.

Source: It was my dad's first official job when he got commissioned in the military (air force). He had to take a psychological evaluation and sign a waiver. This was in Minot, ND.